Common carrier[0] as a concept exists for good reasons, while cloudflare is not technically required to act as one, it just makes perfect sense to not have to make a ambigious content based judgement call for each of your clients.
In many many places, especially in the states there is no such thing as being able to judge with their spending because of monopolies.
No they can't, at least not in America.
Actually they could, but they would lose their common carrier (see a sibling comment) protections.
To the extent that they are not prohibited from doing so, they would also not lose common carrier status if they did. The rules applied to ISPs under Title II -- the FCC's recent Open Internet Order -- do not prohibit blocking unlawful content, only lawful content.